Skip to main content
SpringerLink
Log in

Self-consistent field theory of block copolymers on a general curved surface

  • Regular Article
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract.

In this work, we propose a theoretical framework based on the self-consistent field theory (SCFT) for the study of self-assembling block copolymers on a general curved surface. Relevant numerical algorithms are also developed. To demonstrate the power of the approach, we calculate the self-assembled patterns of diblock copolymers on three distinct curved surfaces with different genus. We specially study the geometrical effects of curved surfaces on the conformation of polymer chains as well as on the pattern formation of block copolymers. By carefully examining the diffusion equation of the propagator on curved surfaces, it is predicted that Gaussian chains are completely unaware of the extrinsic curvature but that they will respond to the intrinsic curvature of the surface. This theoretical assertion is consistent with our SCFT simulations of block copolymers on general curved surfaces.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H.Q. Xiang, K. Shin, T. Kim, S. Moon, T.J. McCarthy, T.P. Russell, J. Polym. Sci. Part B 43, 3377 (2005)

    Article  Google Scholar 

  2. B. Yu, P.C. Sun, T.H. Chen, Q.H. Jin, D.T. Ding, B.H. Li, A.-C. Shi, Phys. Rev. Lett. 96, 138306 (2006)

    Article  ADS  Google Scholar 

  3. Y.Y. Wu, G.S. Cheng, K. Katsov, S.W. Sides, J.F. Wang, J. Tang, G.H. Fredrickson, M. Moskovits, G.D. Stucky, Nat. Mater. 3, 816 (2004)

    Article  ADS  Google Scholar 

  4. H. Xiang, K. Shin, T. Kim, S.I. Moon, T.J. McCarthy, T.P. Russell, Macromolecules 38, 1055 (2005)

    Article  ADS  Google Scholar 

  5. W. Li, R.A. Wickham, Macromolecules 39, 8492 (2006)

    Article  ADS  Google Scholar 

  6. P. Chen, H.J. Liang, A.-C. Shi, Macromolecules 40, 7329 (2007)

    Article  ADS  Google Scholar 

  7. M. Ma, E.L. Thomas, G.C. Rutledge, B. Yu, B.H. Li, Q.H. Jin, D.T. Ding, A.-C. Shi, Macromolecules 43, 3061 (2010)

    Article  ADS  Google Scholar 

  8. P. Chi, Z. Wang, B.H. Li, A.-C. Shi, Langmuir 27, 11683 (2011)

    Article  Google Scholar 

  9. G.H. Griffiths, B. Vorselaars, M.W. Matsen, Macromolecules 44, 3649 (2011)

    Article  ADS  Google Scholar 

  10. B. Yu, B.H. Li, Q.H. Jin, D.T. Ding, A.-C. Shi, Macromolecules 40, 9133 (2007)

    Article  ADS  Google Scholar 

  11. B. Yu, B.H. Li, Q.H. Jin, D.T. Ding, A.-C. Shi, Soft Matter 7, 10227 (2011)

    Article  ADS  Google Scholar 

  12. H. Xu, J.F. Li, H.D. Zhang, Y.L. Yang, Acta Polym. Sin. 5, 548 (2011)

    Article  Google Scholar 

  13. P. Tang, F. Qiu, H.D. Zhang, Y.L. Yang, Phys. Rev. E 72, 016710 (2005)

    Article  ADS  Google Scholar 

  14. J.F. Li, J. Fan, H.D. Zhang, F. Qiu, P. Tang, Y.L. Yang, Eur. Phys. J. E 20, 449 (2006)

    Article  ADS  Google Scholar 

  15. T.L. Chantawansri et al., Phys. Rev. E 75, 031802 (2007)

    Article  ADS  Google Scholar 

  16. G.H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers (Clarendon Press, Oxford, 2006)

  17. M.W. Matsen, M. Schick, Phys. Rev. Lett. 72, 2660 (1994)

    Article  ADS  Google Scholar 

  18. P. Tang, F. Qiu, H.D. Zhang, Y.L. Yang, Phys. Rev. E 69, 031803 (2004)

    Article  ADS  Google Scholar 

  19. P. Tang, F. Qiu, H.D. Zhang, Y.L. Yang, J. Phys. Chem. B 108, 8434 (2004)

    Article  Google Scholar 

  20. Z.J. Guo et al., Phys. Rev. Lett. 101, 028301 (2008)

    Article  ADS  Google Scholar 

  21. F. Drolet, G.H. Fredrickson, Phys. Rev. Lett. 83, 4317 (1999)

    Article  ADS  Google Scholar 

  22. F. Drolet, G.H. Fredrickson, Macromolecules 34, 5317 (2001)

    Article  ADS  Google Scholar 

  23. E. Helfand, J. Chem. Phys. 62, 999 (1975)

    Article  ADS  Google Scholar 

  24. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1989)

  25. J.U. Kim, M.W. Matsen, Macromolecules 41, 4435 (2008)

    Article  ADS  Google Scholar 

  26. B. Vorselaars, J.U. Kim, T.L. Chantawansri, G.H. Fredrickson, M.W. Matsen, Soft Matter 7, 5128 (2011)

    Article  ADS  Google Scholar 

  27. E. Sivaniah, J. Genzer, A. Hexemer, E.J. Kramer, M.L. Xiang, X.F. Li, C.K. Ober, S. Magonov, Macromolecules 41, 9940 (2008)

    Article  ADS  Google Scholar 

  28. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition (World Scientific, Hackensack, NJ, 2006)

  29. M. Berger, A Panoramic View of Riemannian Geometry (Springer-Verlag, Berlin, 2004)

  30. K.A. Brakke, Exp. Math. 1, 141 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  31. J.F. Li, H.D. Zhang, F. Qiu, J. Phys. Chem. B 117, 843 (2013)

    Article  Google Scholar 

  32. M. Meyer, VisMath Proceedings (Berlin, 2002)

  33. J. Sherman, W.J. Morrison, Ann. Math. Stat. 20, 621 (1949)

    Google Scholar 

  34. A. Hexemer, V. Vitelli, E.J. Kramer, G.H. Fredrickson, Phys. Rev. E 76, (2007)

  35. Q. Liang, J.F. Li, P.W. Zhang, J.Z.Y. Chen, J. Chem. Phys. 138, 244910 (2013)

    Article  ADS  Google Scholar 

  36. B. Zhao L. Zhu, Macromolecules 42, 9369 (2009)

    Article  ADS  Google Scholar 

  37. R.P. Carney, G.A. DeVries, C. Dubois, H. Kim, J.Y. Kim C. Singh, P.K. Ghorai, J.B. Tracy, R.L. Stiles, R.W. Murray, S.C. Glotzer, F. Stellacci, J. Am. Chem. Soc. 130, 798 (2008)

    Article  Google Scholar 

  38. W.D. Song, P. Tang, F. Qiu, Y.L. Yang, A.-C. Shi, J. Phys. Chem. B 115, 8390 (2011)

    Article  Google Scholar 

  39. B. Vallet, B. Levy, Comput. Graph. Forum 27, 251 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, 200433, Shanghai, China

    Jianfeng Li, Hongdong Zhang & Feng Qiu

Authors
  1. Jianfeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongdong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianfeng Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, J., Zhang, H. & Qiu, F. Self-consistent field theory of block copolymers on a general curved surface. Eur. Phys. J. E 37, 18 (2014). https://doi.org/10.1140/epje/i2014-14018-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2014-14018-1

Keywords

Search

Navigation